Digital Khalimsky Manifolds

Melin, Erik

doi:10.1007/s10851-008-0114-1

Cited by 10 publications

(6 citation statements)

References 19 publications

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“…In (Z n , κ n ), we say that two distinct points x and y are K-adjacent if y ∈ SN(x) or x ∈ SN(y) [28], where SN(x) is the smallest open neighborhood of the point x. For a point p ∈ Z n , since a permutation of coordinates and a translation by an even vector is a homeomorphism of Z n onto itself [33], we can consider the point p as p = (p i ) i∈ [1,n] Z where there are consecutively k even coordinates from 1 to α coordinates and the other n − α coordinates are odd such as p := (…”

Section: A Development Of An Ha-map Which Is An H-connectedness Preserving Mapmentioning

confidence: 99%

Topological graphs based on a new topology on Zn and its applications

Han

2017

Filomat

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Up to now there is no homotopy for Marcus-Wyse (for short M-) topological spaces. In relation to the development of a homotopy for the category of Marcus-Wyse (for short M-) topological spaces on Z2, we need to generalize the M-topology on Z2 to higher dimensional spaces X ? Zn, n ? 3 [18]. Hence the present paper establishes a new topology on Zn; n 2 N, where N is the set of natural numbers. It is called the generalized Marcus-Wyse (for short H-) topology and is denoted by (Zn, n). Besides, we prove that (Z3, 3) induces only 6- or 18-adjacency relations. Namely, (Z3, 3) does not support a 26-adjacency, which is quite different from the Khalimsky topology for 3D digital spaces. After developing an H-adjacency induced by the connectedness of (Zn; n), the present paper establishes topological graphs based on the H-topology, which is called an HA-space, so that we can establish a category of HA-spaces. By using the H-adjacency, we propose an H-topological graph homomorphism (for short HA-map) and an HA-isomorphism. Besides, we prove that an HA-map (resp. an HA-isomorphism) is broader than an H-continuous map (resp. an Hhomeomorphism) and is an H-connectedness preserving map. Finally, after investigating some properties of an HA-isomorphism, we propose both an HA-retract and an extension problem of an HA-map for studying HA-spaces.

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Section: A Development Of An Ha-map Which Is An H-connectedness Preserving Mapmentioning

confidence: 99%

Topological graphs based on a new topology on Zn and its applications

Han

2017

Filomat

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“…Example 2.1. [13,21] We now characterize A(p), as follows: For a space (X, T n X ) := X we now define the notion of a K-adjacency relation of a point p ∈ X as follows: Definition 2.2. [13] For (X, T n X ) := X put A X (p) := A(p)∩X.…”

Section: Khalimsky Adjacency and Its Propertiesmentioning

confidence: 99%

“…In relation to the establishment of an A-map, we will use the following K-adjacency neighborhood of a point p ∈ X. For a point p ∈ X, we define a KA-neighborhood of p to be the following set [21]…”

Section: Some Properties Of a Ka-map A Ka-isomorphism An A-map And mentioning

confidence: 99%

An extension problem of a connectedness preserving map between Khalimsky spaces

Han

2016

Filomat

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The goal of the present paper is to study an extension problem of a connected preserving (for short, CP-) map between Khalimsky (K-for brevity, if there is no ambiguity) spaces. As a generalization of a K-continuous map, for K-topological spaces the recent paper [13] develops a function sending connected sets to connected ones (for brevity, an A-map: see Definition 3.1 in the present paper). Since this map plays an important role in applied topology including digital topology, digital geometry and mathematical morphology, the present paper studies an extension problem of a CP-map in terms of both an A-retract and an A-isomorphism (see Example 5.2). Since K-topological spaces have been often used for studying digital images, this extension problem can contribute to a certain areas of computer science and mathematical morphology.

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“…Some authors consider digital images as topological spaces rather than as graphs, applying the Khalimsky topology to digital images (see, e.g., [9,11,12,13,14,15]). The Khalimsky topology on Z takes a basic neighborhood of an integer z to be {z} if z is odd; {z − 1, z, z + 1} if z is even.…”

Section: Introductionmentioning

confidence: 99%

Shy maps in topology

Boxer

2018

Topology and its Applications

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Digital Khalimsky Manifolds

Cited by 10 publications

References 19 publications

Topological graphs based on a new topology on Zn and its applications

Topological graphs based on a new topology on Zn and its applications

An extension problem of a connectedness preserving map between Khalimsky spaces

Shy maps in topology

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